Multiplication and Division with Rational Numbers


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1 Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up in Ohio, but they chose Kitty Hawk for its steady winds, soft landings, and privacy. 5.1 Equal Groups Multiplying and Dividing Integers What s My Product or Quotient? Multiplying and Dividing Rational Numbers Properties Schmoperties Simplifying Arithmetic Expressions with Rational Numbers Building a Wright Brothers Flyer Evaluating Expressions with Rational Numbers Repeat or Not? That Is the Question! Exact Decimal Representations of Fractions
2 252 Chapter 5 Multiplication and Division with Rational Numbers
3 Equal Groups Multiplying and Dividing Integers Learning Goals In this lesson, you will: Multiply integers. Divide integers. Pick any positive integer. If the integer is even, divide it by 2. If it is odd, multiply it by 3 and then add 1. Repeat this process with your result. No matter what number you start with, eventually you will have a result of 1. This is known as the Collatz Conjecture a conjecture in mathematics that no one has yet proven or disproven. How do you think it works? 5.1 Multiplying and Dividing Integers 253
4 Problem 1 Multiply Integers When you multiply integers, you can think of multiplication as repeated addition. Consider the expression 3 3 (24). As repeated addition, it means (24) 1 (24) 1 (24) You can think of 3 3 (24) as three groups of (24). 5 ( 4) ( 4) ( 4) Chapter 5 Multiplication and Division with Rational Numbers
5 Here is another example: 4 3 (23). You can think of this as four sets of (23), or (23) 1 (23) 1 (23) 1 (23) ( 3) ( 3) ( 3) ( 3) And here is a third example: (23) 3 (24). You know that 3 3 (24) means three groups of (24) and that 23 means the opposite of 3. So, (23) 3 (24) means the opposite of 3 groups of (24). Opposite of ( 4) Opposite of ( 4) Opposite of ( 4) (+4) (+4) (+4) Multiplying and Dividing Integers 255
6 1. Draw either a number line representation or a twocolor counter model to determine each product. Describe the expression in words. Use the examples if you need help. a b. 2 3 (23) c. (22) 3 3 d. (22) 3 (23) 256 Chapter 5 Multiplication and Division with Rational Numbers
7 2. Complete the table. Expression Description Addition Sentence Product Three groups of (23) (25) (23) 3 (25) 3. Analyze each number sentence What pattern do you notice in the products as the numbers multiplied by 4 decrease? 4. Determine each product. Describe the pattern. a. 4 3 (21) 5 b. 4 3 (22) 5 c. 4 3 (23) Multiplying and Dividing Integers 257
8 5. Write the next three number sentences that extend this pattern How do these products change as the numbers multiplied by 25 decrease? 7. Determine each product. a (21) 5 b (22) 5 c (23) 5 When you multiply by the opposite, you go in the opposite direction! d (24) 5 e. Write the next three number sentences that extend this pattern. 8. What is the sign of the product of two integers when: a. they are both positive? b. they are both negative? c. one is positive and one is negative? d. one is zero? 258 Chapter 5 Multiplication and Division with Rational Numbers
9 9. If you know that the product of two integers is negative, what can you say about the two integers? Give examples. 10. Describe an algorithm that will help you multiply any two integers. 11. Use your algorithm to simplify these expressions. a b (25) 28 3 (27) (27) 26 3 (25) c (24) 23 3 (22) 3 (24) 3 3 (22) (22) (24) Determine the singledigit integers that make each number sentence true. a b c. 3 (29) 5 63 d Multiplying and Dividing Integers 259
10 13. Describe the sign of each product and how you know. a. the product of three negative integers b. the product of four negative integers c. the product of seven negative integers d. the product of ten negative integers 14. What is the sign of the product of any odd number of negative integers? Explain your reasoning. 15. What is the sign of the product of three positive integers and five negative integers? Explain your reasoning. 260 Chapter 5 Multiplication and Division with Rational Numbers
11 Problem 2 Division of Integers When you studied division in elementary school, you learned that multiplication and division were inverse operations. For every multiplication fact, you can write a corresponding division fact. The example shown is a fact family for 4, 5, and 20. Fact Family Similarly, you can write fact families for integer multiplication and division. Examples: (24) (27) (28) (27) (28) (24) What pattern(s) do you notice in each fact family? 2. Write a fact family for 26, 8, and Multiplying and Dividing Integers 261
12 3. Fill in the unknown numbers to make each number sentence true. a (28) 5 b (24) 5 c d e. 4 (28) 5 24 f g. 4 (28) 5 0 h Talk the Talk 1. What is the sign of the quotient of two integers when a. both integers are positive? b. one integer is positive and one integer is negative? c. both integers are negative? d. the dividend is zero? 2. How do the answers to Question 1 compare to the answers to the same questions about the multiplication of two integers? Explain your reasoning. Be prepared to share your solutions and methods. 262 Chapter 5 Multiplication and Division with Rational Numbers
13 What s My Product or Quotient? Multiplying and Dividing Rational Numbers Learning Goals In this lesson, you will: Multiply rational numbers. Divide rational numbers. L ook at these models. The top model shows 6 8, and the bottom model shows 3 8. To determine , you can ask, How many 3 8 go into 6? You can see that the 8 answer, or quotient, is just 6 4 3, or 2. So, if you are dividing two fractions with the same denominators, can you always just divide the numerators to determine the quotient? Try it out and see! 5.2 Multiplying and Dividing Rational Numbers 263
14 Problem 1 From Integer to Rational In this lesson, you will apply what you learned about multiplying and dividing with integers to multiply and divide with rational numbers. 1. Consider this multiplication sentence: ? a. What is the rule for multiplying signed numbers? When you convert a mixed number to an improper fraction, ignore the sign at first. Put it back in when you have finished converting. b. Use the rule to calculate the product. Show your work. 2. Calculate each product and show your work. a b It doesn, t matter what numbers I have. The rules for the signs are the same. c d Chapter 5 Multiplication and Division with Rational Numbers
15 e f Problem 2 And On to Dividing 1. Consider this division sentence: ? a. What is the rule for dividing signed numbers? b. Use the rule to calculate the quotient. Show your work. 2. Calculate each quotient and show your work. a b c d Multiplying and Dividing Rational Numbers 265
16 e f. ( ) Talk the Talk Determine each product or quotient Be prepared to share your solutions and methods. 266 Chapter 5 Multiplication and Division with Rational Numbers
17 Properties Schmoperties Simplifying Arithmetic Expressions with Rational Numbers Learning Goal In this lesson, you will: Simply arithmetic expressions using the number properties and the order of operations. Suppose you didn t know that a negative times a negative is equal to a positive. How could you prove it? One way is to use properties in this case, the Zero Property and the Distributive Property. The Zero Property tells us that any number times 0 is equal to 0, and the Distributive Property tells us that something like 4 3 (2 1 3) is equal to (4 3 2) 1 (4 3 3). We want these properties to be true for negative numbers too. So, start with this: That s the Zero Property. We want that to be true. Now, let s replace the first 0 with an expression that equals 0: 25 3 (5 1 25) 5 0 Using the Distributive Property, we can rewrite that as (25 3 5) 1 ( ) 5 0 Hey, did you hear what Zero said to Eight? "Nice belt." 225 1? 5 0 For the properties to be true, has to equal positive 25! What other number properties do you remember learning about? 5.3 Simplifying Arithmetic Expressions with Rational Numbers 267
18 Problem 1 Properties and Operations 1. For each equation, identify the number property or operation used. Equation Number Property/Operation a b ( ) c. ( ) ( ) d e ( ) 5 ( ( ) ) f g h. ( ) ( ) ( ) i j. (27.02)(23.42) 5 (23.42)(27.02) 268 Chapter 5 Multiplication and Division with Rational Numbers
19 2. For each step of the simplification of the expression, identify the operation or property applied. Number Property/Operation a b. ( ) ( ) c ( ) ( ) 5 28 d ( ) 5 ( ) ( ) 1 ( ) ( ) Simplifying Arithmetic Expressions with Rational Numbers 269
20 3. Supply the next step in each simplification using the operation or property provided. a ( ) Number Property/Operation Commutative Property of Addition Addition Addition b. ( ) Associative Property of Addition Addition Addition c. 25.2( ) 5 Addition Multiplication 270 Chapter 5 Multiplication and Division with Rational Numbers
21 d. 25.1(70 1 3) 5 Distributive Property of Multiplication over Addition Multiplication Addition e. ( ) ( ) 1 ( ) ( ) 5 Distributive Property of Multiplication over Addition Addition Multiplication Problem 2 On Your Own Simplify each expression step by step, listing the property or operation(s) used ( ) 1 5 ( ) 5 Number Property/Operation 5.3 Simplifying Arithmetic Expressions with Rational Numbers 271
22 2. ( ) 1 ( ) 5 Number Property/Operation ( ) 3 ( ) 5 Number Property/Operation Number Property/Operation ( ( 24.3 ) ) 5 Number Property/Operation 6. ( ) ( 6.4 ) 1 ( ) ( ) 5 Number Property/Operation Be prepared to share your solutions and methods. 272 Chapter 5 Multiplication and Division with Rational Numbers
23 Building a Wright Brothers Flyer Evaluating Expressions with Rational Numbers Learning Goals In this lesson, you will: Model a situation with an expression using rational numbers. Evaluate rational expressions. On December 17, 1903, two brothers Orville and Wilbur Wright became the first two people to make a controlled flight in a powered plane. They made four flights that day, the longest covering only 852 feet and lasting just 59 seconds. Human flight progressed amazingly quickly after those first flights. In the year before Orville died, Chuck Yeager had already piloted the first flight that broke the sound barrier! 5.4 Evaluating Expressions with Rational Numbers 273
24 Problem 1 Building a Wright Brothers Flyer In order to build a balsa wood model of the Wright brothers plane, you would need to cut long lengths of wood spindles into shorter lengths for the wing stays, the vertical poles that support and connect the two wings. Each stay for the main wings of the model needs to be cut 3 1 inches long. 4 Show your work and explain your reasoning. 1. If the wood spindles are each 10 inches long, how many stays could you cut from one spindle? 2. How many inches of the spindle would be left over? 274 Chapter 5 Multiplication and Division with Rational Numbers
25 3. If the wood spindles are each 12 inches long, how many stays could you cut from one spindle? 4. How many inches of the spindle would be left over? You also need to cut vertical stays for the smaller wing that are each 1 5 inches long If the wood spindles are each 10 inches long, how many of these stays could you cut from one spindle? 5.4 Evaluating Expressions with Rational Numbers 275
26 6. How many inches of the spindle would be left over? 7. If the wood spindles are each 12 inches long, how many stays could you cut from one spindle? 276 Chapter 5 Multiplication and Division with Rational Numbers
27 8. How many inches of the spindle would be left over? 9. Which length of spindle should be used to cut each of the different stays so that there is the least amount wasted? 5.4 Evaluating Expressions with Rational Numbers 277
28 Problem 2 Building a Wright Brothers Flyer Redux Remember, a stay is inch. There are longer spindles that measure 36 inches. 1. How much of a 36inchlong spindle would be left over if you cut one of the stays from it? Show your work and explain your reasoning. 2. How much of this spindle would be left over if you cut two of the stays from it? 3. Define variables for the number of 3 1 inch stays and the amount of the 36inchlong 4 spindle that is left over. 278 Chapter 5 Multiplication and Division with Rational Numbers
29 4. Write an equation for the relationship between these variables. 5. Use your equation to calculate the amount of the spindle left over after cutting 10 stays. 6. Use your equation to calculate the amount of the spindle left over after cutting 13 stays. 5.4 Evaluating Expressions with Rational Numbers 279
30 Problem 3 Evaluating Expressions 1. Evaluate the expression ( ) v for: a. v 5 25 To evaluate an expression, substitute the values for the variables and then perform the operations. b. v 5 3 c. v d. v Chapter 5 Multiplication and Division with Rational Numbers
31 2. Evaluate the expression ( ) x for: a. x b. x Be prepared to share your solutions and methods. 5.4 Evaluating Expressions with Rational Numbers 281
32 282 Chapter 5 Multiplication and Division with Rational Numbers
33 Repeat or Not? That Is the Question! Exact Decimal Representations of Fractions Learning Goals In this lesson, you will: Use decimals and fractions to evaluate arithmetic expressions. Convert fractions to decimals. Represent fractions as repeating decimals. Key Terms terminating decimals nonterminating decimals repeating decimals nonrepeating decimals bar notation Sometimes calculating an exact answer is very important. For example, making sure that all the parts of an airplane fit exactly is very important to keep the plane in the air. Can you think of other examples where very exact answers are necessary? 5.5 Exact Decimal Representations of Fractions 283
34 Problem 1 Not More Homework! Jayme was complaining to her brother about having to do homework problems with fractions like this: ( ) ? Jayme said, I have to find the least common denominator, convert the fractions to equivalent fractions with the least common denominator, and then calculate the answer! Her brother said, Whoa! Why don t you just use decimals? 1. Calculate the answer using Jayme s method. 2. Convert each mixed number to a decimal and calculate the sum. 3. In this case, which method do you think works best? Jayme said: That s okay for that problem, but what about this next one? ( ) 1 ( ) Chapter 5 Multiplication and Division with Rational Numbers
35 4. Calculate the answer using Jayme s method. 5. Will Jayme s brother s method work for the second problem? Why or why not? Problem 2 Analyzing Decimals 1. Convert each fraction to a decimal. a. 11 b c. 27 d Exact Decimal Representations of Fractions 285
36 e. 7 f g. 7 h i. 3 j Decimals can be classified in four different ways: terminating, nonterminating, repeating, or nonrepeating. A terminating decimal has a finite number of digits, meaning that the decimal will end, or terminate. A nonterminating decimal is a decimal that continues without end. A repeating decimal is a decimal in which a digit, or a group of digits, repeat(s) without end. A nonrepeating decimal neither terminates nor repeats. Bar notation is used for repeating decimals. Consider the example shown. The sequence repeats. The numbers that lie underneath the bar are those numbers that repeat The bar is called a vinculum. 286 Chapter 5 Multiplication and Division with Rational Numbers
37 2. Classify each decimal in Question 1, parts (a) through ( j) as terminating, nonterminating, repeating, or nonrepeating. If the decimal repeats, rewrite it using bar notation. 3. Can all fractions be represented as either terminating or repeating decimals? Write some examples to explain your answer. 4. Complete the graphic organizer. Describe each decimal in words. Show examples. Be prepared to share your solutions and methods. 5.5 Exact Decimal Representations of Fractions 287
38 Terminating NonTerminating Decimals p is a wellknown nonrepeating decimal. You will learn more when you study circles later in this course. Repeating NonRepeating 288 Chapter 5 Multiplication and Division with Rational Numbers
39 Chapter 5 Summary Key Terms terminating decimals (5.5) nonterminating decimals (5.5) repeating decimals (5.5) nonrepeating decimals (5.5) bar notation (5.5) Multiplying Integers When multiplying integers, multiplication can be thought of as repeated addition. Twocolor counter models and number lines can be used to represent multiplication of integers. Example Consider the expression 3 3 (23). As repeated addition, it means (23) 1 (23) 1 (23) The expression 3 3 (23) can be thought of as three groups of (23). Dividing complicated tasks, or problems, into simpler pieces can help your brain better learn and understand. Try it! = (3) (3) (3) Chapter 5 Summary 289
40 Dividing Integers Multiplication and division are inverse operations. For every multiplication fact, there is a corresponding division fact. Example This is a fact family for 3, 8, and Determining the Sign of a Product or Quotient The sign of a product or quotient of two integers depends on the signs of the two integers being multiplied or divided. The product or quotient will be positive when both integers have the same sign. The product or quotient will be negative when one integer is positive and the other is negative. Example Notice the sign of each product or quotient. he Multiplying and Dividing Rational Numbers The rules used to determine the sign of a product or quotient of two integers also apply when multiplying and dividing rational numbers. Example The product or quotient of each are shown following the rules for determining the sign for each Chapter 5 Multiplication and Division with Rational Numbers
41 Simplifying Expressions with Rational Numbers When simplifying arithmetic expressions involving rational numbers, it is often helpful to identify and use the number properties or operations that make the simplification easier. Example The steps for simplifying the expression are shown ( ) ( ) 5 Number Property/Operation ( ) 5 Distributive Property of Multiplication over Addition ( 8 ) 5 Addition 22 Multiplication Evaluating Expressions with Rational Numbers To evaluate an expression containing variables, substitute the values for the variables and then perform the necessary operations. Example The evaluation of the expression ( m ) when m 5 7 is shown ( ) ( 4 5 ) ( 4 5 ) ( 1 1 ) 5 7 Chapter 5 Summary 291
42 Representing Fractions as Decimals To convert a fraction to a decimal, divide the numerator of the fraction by the denominator. A terminating decimal has a finite number of digits, meaning that the decimal will end or terminate. A nonterminating decimal is a decimal that continues without end. A repeating decimal is a decimal in which a digit, or a group of digits, repeats without end. When writing a repeating decimal, bar notation is used to indicate the digits that repeat. A nonrepeating decimal neither terminates nor repeats. Example The fraction 3 is a terminating decimal. The decimal equivalent of 3 is The fraction 2 is a nonterminating, repeating decimal. The decimal equivalent of is Using bar notation, 2 is written as Chapter 5 Multiplication and Division with Rational Numbers
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